For real numbers $t,$ the point
\[(x,y) = \left( e^t + e^{-t}, 3 (e^t - e^{-t}) \right)\]is plotted.  All the plotted points lie on what kind of curve?

(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola

Enter the letter of the correct option.

Note: $e$ is a mathematical constant that is approximately $2.71828.$
Solution: We have that
\[x^2 = (e^t + e^{-t})^2 = e^{2t} + 2 + e^{-2t},\]and
\begin{align*}
\frac{y^2}{9} &= (e^t - e^{-t})^2 \\
&= e^{2t} - 2 + e^{-2t}.
\end{align*}Then
\[x^2 - \frac{y^2}{9} = 4,\]so
\[\frac{x^2}{4} - \frac{y^2}{36} = 1.\]Thus, all plotted points lie on a hyperbola.  The answer is $\boxed{\text{(E)}}.$